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A Polynomial Approach for Analysis and Optimal

Control of Switched Nonlinear Systems

Eduardo Mojica-Nava

To cite this version:

Eduardo Mojica-Nava. A Polynomial Approach for Analysis and Optimal Control of Switched Non-linear Systems. Automatic. École Centrale de Nantes, 2009. English. �tel-02723110�

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Ecole Centrale de Nantes - Universidad de Los Andes

´

Ecole Doctorale

Sciences et Technologies

de l’Information et de Mathematiques

Ann´ee 2009 Provisional Version September 20 2009

Th`ese de Doctorat

Diplˆome d´elivr´e par L’ ´Ecole Centrale de Nantes

Sp´ecialit´e : Automatique et Traitement du Signal Pr´esent´ee et soutenue publiquement par:

Eduardo Mojica-Nava

le 22 septembre 2009 `

a l’´Ecole des Mines de Nantes, Nantes Titre

A Polynomial Approach for Analysis and Optimal

Control of Switched Nonlinear Systems

Jury

Rapporteurs : Peter Caines – Professor, McGill University (Montreal, Canada)

Didier Henrion – CNRS Researcher, LAAS (Toulouse, France)

Membre invit´e du jury : Pierre Riedinger – Maˆıtre de Conf´erences, CRAN-INPL (Nancy, France)

Examinateurs :

Directeur France: Jean Jacques Loiseau – Directeur de Recherche CNRS, IRRCyN (Nantes, France)

Directeur Colombie: Alain Gauthier – Professor, Univ. de los Andes (Bogot´a, Colombie)

Co-encadrant France: Naly Rakoto-Ravalontsalama – Maˆıtre Assistant, EMN (Nantes, France)

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Para mis padres,

Alirio y Cecilia,

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ACKNOWLEDGEMENTS

First and foremost, I would like to express deep gratitude towards my advisors, Alain Gauthier, Naly Rakoto, Jean-Jacques Loiseau, and Nicanor Quijano. Alain, for his guidance and support all these years; I will be forever thankful for this opportunity. Naly, for providing a unique working environment that I mostly enjoyed and also for his encouragement during the time at EMN. Jean-Jacques, for letting me be part of the research group ACSED under his direction, and also for his many interesting advices. At last but not the least, my greatest appreciation goes to my closest advisor, Nicanor. His advice and friendship lead me through the life of a graduate student; Nick made my Uniandes experience terrific, creating numerous opportunities for me to learn and grow as a researcher.

I also thank the other members of my doctoral committee: Peter Caines, Didier Henrion and Pierre Riedinger, for their encouragement, generosity, and insightful discussions. Furthermore, I am grateful to Rene Meziat for our long conversations, many of which found their way into the pages of this thesis.

To all my friends at Uniandes: Andr´es Pantoja, Iv´an mu˜noz, and the GIAP group

gang, thank you for the Caf´e and Coffee evenings—you have been great. Furthermore,

I thank my closest friends at EMN, Yonatan and Chio, for their noble hearts. I also

want to mention Roberto Bustamante and ´Angela Cadena, professors at Uniandes,

for their good humor and inspiration.

Finally, I am forever grateful to my parents, Alirio and Cecilia, and the rest of my family for their constant encouragement, confidence, friendship, and unconditional love for all these years. And also, I want to express my endless love to Carito, “mi mitis”, who even though was for almost a year 8,960 km away, always waited for me.

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Une Approche polynomiale pour l’analyse et la commande

optimale des syst`

emes non-lin´

eaires `

a commutation

esum´

e :

Dans cette th`ese nous ´etudions comment la g´eom´etrie semi-alg´ebrique

convexe et l’optimisation polynˆome globale peuvent ˆetre employ´ees pour analyser et

concevoir les syst`emes non lin´eaires `a commutations. Pour traiter l’analyse de

sta-bilit´e des syst`emes non-lin´eaires `a commutations on montre que la transformation

du probl`eme original `a commutations vers un syst`eme polynˆomial continu nous

per-met d’employer l’in´egalit´e de dissipation pour les syst`emes polynˆomiaux. Avec cette

m´ethode et d’un point de vue th´eorique, nous fournissons une mani`ere alternative

de rechercher une fonction commune de Lyapunov pour les syst`emes non lin´eaires `a

commutations.

L’id´ee principale derri`ere l’approche propos´ee est d’inclure dans l’analyse

fonction-nelle les contraintes cach´ees. Nous devons v´erifier le d´efinition semi-n´egative de dV /dt

en ce qui concerne l’ensemble de contraintes. Pour cela, nous employons l’id´ee de la

p´enalisation utilis´ee dans la th´eorie d’optimisation avec contraintes. Une fonction

λ(x, s) est introduite et elle peut ˆetre interpr´et´ee comme fonction de p´enalisation ou

multiplicateur de Lagrange. Cette id´ee est bas´ee sur des r´esultats pour les syst´emes

de commande contraints, o´u nous pouvons employer le concept d’in´egalit´e de

dissipa-tion utilisant des foncdissipa-tions de stockage et des taux d’approvisionnement. Pour cela,

nous employons l’id´ee de la p´enalisation utilis´ee dans la th´eorie d’optimisation avec

des contraintes. Ainsi nous ´etendons alors les r´esultats ´a une classe plus g´en´erale

des syst´emes commut´es, ceux mod´elis´es par des fonctions ´el´ementaires. Cette classe

de fonctions provient des d´eriv´es symboliques explicites, telles que l’exponentielle, le

logarithme, les fonctions trigonom´etriques, et les fonctions hyperboliques. Pour ce

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par la repr´esentation ´equivalente dans un syst´eme sous la forme polynˆomiale et puis,

nous employons les r´esultats de la section pr´ec´edente pour l’analyse de la stabilit´e.

En plus de l’analyse de stabilit´e, des probl`emes de commande optimale pour les

syst`emes non-lin´eaires commut´es sont ´egalement ´etudi´es. Nous proposons une

ap-proche alternative pour r´esoudre le probl`eme de commande optimale pour un syst`eme

non lin´eaire autonome `a commutations, bas´e sur le principe de maximum g´en´eralis´e

(GMP). L’essentiel de cette m´ethode est la transformation d’un probl`eme de

com-mande optimale non-lin´eaire et non-convexe, c’est-´a-dire, le syst`eme commut´e, en

un probl`eme de commande optimale ´equivalent avec la structure lin´eaire et convexe,

qui permet d’obtenir une formulation convexe ´equivalente plus appropri´ee pour ˆetre

r´esolue par un calcul num´erique plus efficace. En cons´equence, nous proposons de

con-vexifier les variables d’´etat et de commande au moyen de la m´ethode des moments

afin d’obtenir des programmes SDP. Une g´en´eralisation pour r´esoudre le probl`eme

de commande optimale des syst`emes commut´es non-lin´eaires est ´etudi´ee `a partir du

processus r´e´ecrit.

En conclusion, nous ´etudions l’application industrielle obtenue par une

approxi-mation lin´eaire par morceaux de la croissance cellulaire non-lin´eaire en utilisant des

fonctions canoniques lin´eaires orthonormales. Elle est command´ee par une strat´egie

de ¡¡probing control¿¿. Nous traitons les cellules mammif`eres BHK (rein de b´eb´e

hamster) dans un bio-r´eacteur. Les r´esultats de simulation prouvent que cette

approximation lin´eaire par morceaux est bien adapt´ee pour model´eliser une telle

dynamique non-lin´eaire.

Mots cl´

es :

Optimisation convexe, commande optimale, syst`emes

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A Polynomial Approach for Analysis and Optimal Control of

Switched Nonlinear Systems

Abstract:

In this dissertation we investigate how convex semialgebraic geometry and global polynomial optimization can be used to analyze and to design switched nonlinear systems. To deal with stability analysis of switched nonlinear systems it is shown that the representation of the original switched problem into a continuous polynomial system allows us to use the dissipation inequality for polynomial systems. With this method, and from a theoretical point of view, we provide an alternative way to search for a common Lyapunov function for switched nonlinear systems.

The main idea behind the proposed approach is to include in the system analysis

the hidden constraints. We need to check negative semidefinitness of ˙V with respect

to the constrained set. In order to do that, we use the idea of penalization used in optimization theory with constraints. For that, we use a function λ(x, s), which can be interpreted as a penalization function or a Lagrange multiplier. This idea is based on some results for constrained control systems, where we can use the dissipation inequality concept using storage functions and supply rates. We then extend the results to a more general class of switched systems, those modeled by elementary and nested elementary functions. This class of functions is related to explicit symbolic derivatives, such as exponential, logarithm, power-law, trigonometric, and hyperbolic functions. For this aim, we transform, using a recasting process, the system obtained by the equivalent representation in a system with polynomial form, and then we use the results of the previous section for stability analysis.

Besides stability analysis, optimal control problems for switched nonlinear systems are also investigated. We propose an alternative approach for solving effectively the optimal control problem for an autonomous nonlinear switched system based on the

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Generalized Maximum Principle (GMP). The essence of this method is the transfor-mation of a nonlinear, non-convex optimal control problem, i.e., the switched system, into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high-performance numerical computing. Consequently, we propose to convexify the state and control variables by means of the method of moments obtaining SDP programs. A generalization to solve the optimal control problem of nonlinear switched systems based on the recasting process is investigated then.

Finally, we concentrate in the industrial application obtaining a piecewise-linear approximation of nonlinear cellular growth using orthonormal canonical piecewise linear functions, which is tested by a probing control strategy for the feed rate. We deal with the mammalian cells BHK (Baby Hamster Kidney) in bioreactor in batch, fed-batch, and continuous mode operation. Simulation results show that this piecewise linear approximation is well suited for modeling such nonlinear dynamics.

Keywords:

Convex optimization, optimal Control, polynomial systems, switched systems, stability analysis

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TABLE OF CONTENTS

DEDICATION . . . i ACKNOWLEDGEMENTS . . . ii LIST OF TABLES . . . x LIST OF FIGURES . . . xi I INTRODUCTION . . . 1

1.1 Introductory Remarks and Motivation . . . 1

1.2 Contributions, Literature Review, and Outline . . . 3

1.2.1 For Stability Analysis . . . 3

1.2.2 For the Optimal Control Problem . . . 5

1.2.3 For the Piecewise Linear Model and Control of a Bioreactor 9 II A POLYNOMIAL APPROACH FOR STABILITY ANALYSIS OF SWITCHED SYSTEMS . . . 13

2.1 Definitions and Preliminaries . . . 14

2.1.1 Basic Concepts . . . 14

2.1.2 Stability Analysis under Arbitrary Switching and Dissipativity 15 2.2 An Equivalent Polynomial Representation . . . 19

2.3 Results in Stability Analysis for Polynomial Constrained Dynamical Systems . . . 21

2.3.1 The Sum of Squares Decomposition . . . 24

2.3.2 Numerical Example of a Polynomial Switched System . . . 26

2.4 A Generalization for Nonlinear Switched Systems . . . 28

2.4.1 The Recasting Process for Stability Analysis . . . 28

2.4.2 Example of a Non-Polynomial Switched System . . . 31

III ON OPTIMAL CONTROL OF SWITCHED SYSTEMS USING A POLY-NOMIAL APPROACH . . . 34

3.1 Definitions and Preliminaries . . . 35

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3.1.2 Maximum Principle and Necessary Conditions . . . 36

3.1.3 Relaxation and Young Measures . . . 39

3.2 An Equivalent Polynomial Optimal Control Problem . . . 42

3.2.1 Equivalent Representations . . . 42

3.2.2 Equivalent Optimal Control Problem . . . 43

3.3 Relaxation of the Equivalent Optimal Polynomial Problem . . . 46

3.3.1 SDP Relaxation of the Optimal Control Problem . . . 51

3.3.2 Switched Optimization Algorithm . . . 55

3.3.3 Numerical Example: Artstein’s Circle . . . 56

3.4 Extension Results to More General Nonlinear Optimal Control Prob-lems . . . 59

3.4.1 The Recasting Process . . . 61

3.4.2 SDP Relaxation . . . 64

3.4.3 Numerical Example: Swinging up a Pendulum . . . 66

IV PIECEWISE-LINEAR APPROACH TO NONLINEAR CELLULAR GROWTH CONTROL . . . 71

4.1 Process Description . . . 72

4.1.1 Nonlinear Model . . . 73

4.2 The Biological CPWL model . . . 76

4.2.1 Orthonormal Canonical Piecewise Linear Functions . . . 76

4.2.2 Analysis of CPWL Approximation: Error Estimation . . . . 81

4.2.3 Cellular Growth CPWL Model . . . 82

4.3 Simulation Results . . . 83

4.3.1 Model Simulation Results: Nonlinear vs CPWL . . . 83

4.3.2 Transient Analysis of Cells Concentration . . . 86

4.4 Probing Feed Controller . . . 86

4.4.1 Feedback Algorithm . . . 88

V CONCLUSIONS AND FUTURE WORK . . . 92

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5.2 Future Research Directions . . . 93

APPENDIX A MATHEMATICAL BACKGROUND . . . 95

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LIST OF TABLES

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LIST OF FIGURES

1 Switching between stable systems that are becoming stable . . . 15

2 Switching between stable systems that are becoming unstable . . . . 16

3 Phase plane for two different initial conditions . . . 57

4 States, co-states, and switching signal for the Arstein’s circle example 60 5 Phase plane of the system response for the Arstein’s circle example . 61 6 States and switching signal for the pendulum example . . . 70

7 A cell bioreactor in a feed-batch operation mode . . . 75

8 Schematic of the Bioreactor with CPWL model . . . 77

9 Simplicial partition and grid size . . . 79

10 Comparison of the system response computed with the nonlinear and CPWL models to a dilution rate u = 0.0 . . . . 84

11 Comparison of the system response computed with nonlinear and CPWL models to the dilution u = 0.031 . . . . 85

12 Transient response of cells and glucose concentration for u=0.036 for different start-up conditions . . . 87

13 Flow diagram of the probing control algorithm ([7]) . . . 89

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CHAPTER I

INTRODUCTION

1.1

Introductory Remarks and Motivation

Hybrid systems arise in a wide variety of practical systems. We start pointing out that the term hybrid system can be understood from several point of views. From the technological point of view, systems that contain analog and digital components, systems that comprise part of different physical natures such as biological, chemical, electrical, electronic, hydraulic and mechanical ones, and more generally, settings that involve the use of computers for control proposes are termed hybrid systems. From a mathematical modeling point of view, systems described in different forms, such as algebraic equations, difference equations, ordinary differential equations, logical equations, and partial differential equations, are hybrid systems. In the opening article of the European Journal of Control [21] (1995), hybrid systems take a more prominent position. Hybrid systems are mentioned among the major open problems in systems and control theory by several respondents, including Lennart Ljung, Peter Caines, and Pravin Varaiya. The thrust of the thinking on the subject can be seen from Vidyasagar’s remark:

Another interesting question is: ‘How can one combine

differen-tial/difference equations with logical switches so as to enhance perfor-mance?’ In some sense, this is the central question of intelligent control. It seems therefore that by the mid-nineties, hybrid systems have been clearly identified as a major new research area for systems and control theory, and they still constitute a relatively new and very active area of current research (e.g., [9], [72], [1], [2], [3], [45], [111], [4], [5], [6]). In spite of this current interest in hybrid systems, they have

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been with us at least since the days of the relay. The earliest reference we know of is the work of Witsenhausen from MIT, who formulated a class of hybrid-state continuous-time dynamic systems and investigated an optimal control problem [106]. It is worthwhile mention in that there are various models for hybrid systems; due to its inherently interdisciplinary nature, the field has attracted the attention of people with diverse backgrounds, primarily computer scientists, applied mathematicians, and engineers [54], [99], [55]. However, we consider continuous-time systems with discrete switching events, which consist of several subsystems and a switching law that determines the switching times and mode transitions. Such systems are called switched systems and can be viewed as higher-level abstraction of hybrid systems [54]. Switched system modeling of any real-process dealing with physical variables are in agreement with the time-continuous and uniqueness principle, i.e., the value of every physical variable changes only continuously in time through every intermediate value (initial and final), and by possessing a unique value at a specific time and space. Any synthesized control should be uniquely defined and continuous in time. Recent efforts in switched systems research have been typically focused on the analysis of dynamic behaviors, such as stability, controllability and observability, and optimal control, among others (e.g., [99], [56], [54]).

In this dissertation we deal with three different problems of switched systems:

• stability analysis under arbitrary switching, • optimal control problem, and

• piecewise linear model and control of a bioreactor.

The first two problems are related by means of an equivalent polynomial represen-tation. The third problem is an industrial application which uses a class of switched system when the switching law is decided by the partition of the state space. We

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present a brief introduction of each subject of the dissertation with its corresponding chapter.

1.2

Contributions, Literature Review, and Outline

1.2.1 For Stability Analysis

We deal with the stability analysis of switched non-linear systems under arbitrary switching. Most of the efforts in switched systems research have been typically fo-cused on the analysis of dynamical behavior with respect to switching signals. Sev-eral methods have been proposed for stability analysis (see [54], [56], and references therein), but most of them have been focused on switched linear systems. Stability analysis under arbitrary switching is a fundamental problem for the analysis and de-sign of switched systems. For this problem, it is necessary that all the subsystems are asymptotically stable. However, in general, the above stability condition is not sufficient to guarantee stability for the switched system under arbitrary switching. It is well known that if there exists a common Lyapunov function for all the subsystems, then the stability of the switched system is guaranteed under arbitrary switching. Pre-vious attempts of general constructions of a common Lyapunov function for switched nonlinear systems have been presented in [30] and [58], using converse Lyapunov the-orems. Also in [103], a construction of a common Lyapunov function is presented for the case when the individual systems are handled sequentially rather than si-multaneously for a family of pairwise commuting systems. These methodologies are presented in a very general framework, and even though they are mathematically sound, they are too restrictive from a computational point of view, mainly because it is usually hard to check for the set of necessary conditions for a common function over all the subsystems (which might not exist). Also, these constructions are usually iterative, which involves running backward in time for all possible switching signals, being prohibitive when the number of modes increase.

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The main contribution of Chapter 2 is twofold. First, we present a reformulation of the switched system as a differential continuous system on a constraint manifold. This representation opens several possibilities of analysis and design of switching systems in a consistent way, and also with numerical efficiency [69], [70], which is possible thanks to some tools developed in the last decade for polynomial differential-algebraic equations analysis [32], [75], [44]. The second contribution is to show an alternative method to search for a common Lyapunov function for switched systems with an efficient numerical method, using results from stability analysis of polynomial systems based on dissipativity theory [31], [70]. We propose a methodology to con-struct common Lyapunov functions for switched non-linear systems, which provides a less conservative test for proving stability under arbitrary switching. It has been mentioned in [82] that the sum of squares decomposition, presented only for switched polynomial systems, can sometimes be made for a system with a non-polynomial vector fields. These cases, where possible, are restricted to subsystems, which after the rendering in polynomial forms using auxiliary variables, preserve all the same dimensions. However, to our knowledge this has not been shown in the literature. The methodology that we propose does not have the dimentionality limitation men-tioned above. In a previous work [70], we have presented the method only for the case when all the subsystems are in a polynomial form. Later, we extend some pre-liminary results to a more general non-linear case, and a representative example is presented to show the effectiveness of the methodology by reliable and efficient nu-merical methods. Basically, this theory is based in terms of an inequality involving a generalized system power input, or supply rate, and a generalized energy function, or storage function [105]. The interpretation of this storage function establishes the connection between Lyapunov stability and dissipativity. Stability problems can be solved once the dissipativity property is assured, and the storage function becomes a Lyapunov function, which can be used to construct Lyapunov functions for nonlinear

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dynamical systems. As for a common Lyapunov function, a single storage function for all subsystems is usually difficult to find or may not exist (computational problems arise when a common function needs to be found). However, thanks to the compu-tational tools that have been developed lately, we are able to use dissipativity theory with efficient numerical methods to establish a common Lyapunov function for the equivalent polynomial system.

Alternatively, the authors in [54] propose a Lie algebraic condition for switched LTI systems, which is based on the solvability of the Lie algebra generated by the set of state matrices. The Lie algebraic condition is also extended to switched nonlinear systems to obtain local stability results based on Lyapunov’s first method. Most recently global stability properties for switched nonlinear systems are presented in [103], [59], [60], and a Lie algebraic global stability criterion is derived, based on Lie brackets of the nonlinear vector fields. This sort of analysis based on algebraic conditions and Lie algebra are not considered in this work.

1.2.2 For the Optimal Control Problem

1.2.2.1 A Brief Literature Review for Optimal Control of Hybrid Systems

The earliest reference we know of optimal control for hybrid systems is the seminal paper of Witsenhausen [106](1966), where an optimal terminal constraint problem was considered on his hybrid systems model. Later in [94], an optimal control for switching systems was presented, followed by the influential work [24], and [23] where the authors compared several algorithms for optimal control and discuss general con-ditions for the existence of optimal control laws. Eventually, necessary optimality conditions for hybrid systems were derived using general versions of the maximum principle [100], [88], [86] and more recently in [37], and in particular for switching sys-tems in [19] and [104], where the switched system was embedded into a larger family of systems and the optimization problem was formulated. In some recent papers, [97] and [29], we can find some work related with embedding approach for the linear case.

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However, they do not provide further insights on how to find the optimal switching strategy. For general hybrid systems, with nonlinear dynamics in each location and with autonomous and controlled switchings, necessary optimality conditions were re-cently presented in [95], and using these conditions, algorithms based on the hybrid maximum principle were derived. An approach based on the parameterization of the switching instants and the differentiation of the cost function was presented in [107], [108], [109]. The algorithm proposed is based on a two-stage optimization problem. However, the method encounters major computational difficulties when the number of available switches increases.

Lincoln and Rantzer [57] presented the method dubbed relaxing dynamic

program-ming to approximate hybrid optimal control laws and to compute lower and upper

bounds of the optimal cost, while the case of piecewise-affine systems was discussed by Rantzer and Johansson [84]. For determining the optimal feedback control law, these techniques require the discretization of the state space in order to solve the cor-responding Hamilton-Jacobi-Bellman equations that make it intractable numerically. For discrete-time linear hybrid systems, Bemporad and Morari introduced a hy-brid modeling framework that, in particular, handles both internal switches (i.e., caused by the state reaching a particular boundary) and controllable switches [16]. The authors also showed how mixed-integer quadratic programming (MIQP) can be used to determine optimal control sequences. On the other hand, it is generally perceived that the best numerical methods available for hybrid optimal control prob-lems involve mixed integer programming (MIP). While great progress has been made in recent years in improving these methods, the MIP is an NP-hard problem, so scalability is problematic [104]. Bemporad and Morari have worked in model pre-dictive control with different problems (e.g., constrained finite time optimal control (CFTOC), constrained infinite time optimal control (CITOC)) [17]. In [71] Morari and Baric presented the recent developments in control constrained hybrid systems,

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in which the control paradigm is focused on MPC, with the emphasis on explicit so-lution. Nonlinear parametric optimization using cylindrical algebraic decomposition is presented in [34], [35].

For cases where online optimization is not viable, Seatzu et al. proposed a mul-tiparametric programming for solving in state-feedback form the infinite-time hybrid optimal control, by showing that the resulting optimal control law is piecewise affine [93]. They considered the optimal control of continuous-time switched affine systems with a piecewise-quadratic cost function by two methods: i) the so-called master-slave procedure (MSP), and ii) the switching table procedure (STP). The drawback of all these approaches is that they take a lot of computing time.

Focusing on real-time application, Egerstedt et al. [33] considered an optimal control problem for switched dynamical systems, where the objective is to minimize a cost functional defined on the state, and where the control variable consists of the switching times. A gradient-descent algorithm is proposed based on an especially

simple form of the gradient of the cost functional. In [12] and [22] the authors

deal with the problems of mode-switching with an unknown initial state and the construction of a surface for optimality. Such systems change modes whenever the state intersects certain surfaces that are defined in the state space.

In [52] a control parameterization enhancing transform is presented with pre-specified order of the sequence of subsystems, where the switching instants are in-cluded in the cost functional. Both the switching instants and the control function are to be chosen in a way that the cost functional is minimized. In [11], [8] an algorithm based on strong variations to handle constraints on both locations and switching in-stants is proposed for switched nonlinear systems. With the advent of differential inclusion theory, some results using this technique are presented by Vinter in [36], in which the continuous subsystems are modeled as differential inclusions. A distinctive feature of the analysis is that it permits an infinite set of discrete states.

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On the other hand, the H control problem for nonlinear switched systems is addressed in [114] where, based on multiple Lyapunov functions, a sufficient condition for the problem to be solved is derived in terms of partial differential inequalities. The continuous controllers for each subsystem and the switching law are simultaneously designed.

1.2.2.2 Contributions on Optimal Control of Switched Systems

The main contribution of Chapter 3 is an alternative approach to solve effectively the optimal control problem for an autonomous nonlinear switched system based on the Generalized Maximum Principle (GMP) introduced in [110], and later used in [78] and [73] to establish existence conditions for an infinite-dimensional linear program over a space of measure. At a first stage, we focus our analysis on vector fields and running costs that are of polynomial form. However, it is well known that functions called nested elementary functions can be recasted exactly in polynomial systems with a larger state dimension. We will therefore use the fact that all system data are polynomial after the recasting process. We will then apply the Theory of Moments, a method previously introduced for global optimization in [46], [47], [49], [61], and for variational calculus in [63] and with some previous results recently introduced for optimal control problems in [51, 49, 80, 62, 40, 50, 69]. The moment approach for global polynomial optimization based on semidefinite programming (SDP) is consis-tent, as it simplifies and/or has better convergence properties when solving convex problems [48]. This approach works properly when the control variable (i.e., the switching signal), and the state variables can be expressed as polynomials. The es-sential of this method is the transformation of a nonlinear, non-convex optimal control problem (i.e., the switched system), into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formu-lation more appropriate to be solved by high-performance numerical computing. In

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other words, we transform a given controllable switched system into a controllable continuous polynomial system with a linear and convex structure. Should we use a nonlinear, non-convex form of the control variable, we would not be able to use the Hamilton equations of the maximum principle, and nonlinear mathematical program-ming techniques. That would entail severe difficulties, either in the integration of the Hamilton equations or in the search method of any numerical optimization algorithm. Consequently, we propose to convexify the state and control variables by using the method of moments in the polynomial expression in order to deal with this kind of problems. Finally, we use our previous work, where we have limited our analysis to vector fields and running costs of polynomial form, to extend the result to a more general, nonlinear switched systems by means of the ideas introduced in [92] that help us to cope with these non-polynomial terms, which are based on the recasting process of a specific kind of non-polynomial functions.

1.2.3 For the Piecewise Linear Model and Control of a Bioreactor

Mammalian cells of Baby Hamster Kidney (BHK) are used in the production of the vaccine against the foot-and-mouth disease. These cells display multiple steady states with widely varying concentrations of cell mass, desired products, and also waste metabolites [74], [65], [64]. It means that for identical input conditions to a fed-batch reactor, the outlet conditions change depending on how the culture is made fed-batch. These multiple states are manifestations of the complex interaction between cells and their environment. What make this process difficult is the additional level of complexity present in biological systems because of the genetic information in living cells. Several nonlinear models have been developed for mammalian cells (see [64] and references therein), but most of them arise in computational problems. Usually, for nonlinear models from the point of view of control design, details about intracellular metabolism are omitted. The models are based on macroscopic mass balance, and

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include only the more relevant biological reactions. In spite of these attempts to find simple but useful nonlinear models, the modeling of the reaction kinetics are generally represented by rational functions of the state and numerous studies have shown that modeling of the kinetics is a very difficult task [14].

The peculiar features of mammalian cells growth in a fed-batch operating condi-tion are addressed. The task of the controller is to determine, at every instant, the best feed substrate, using the compilation of information online from the sensor. The determination of an optimal strategy of feed substrate using the nonlinear modeling even if the kinetics are known, is not a straightforward matter and is often further complicated by the presence of constraints imposed on the state variables [14]. All of these difficulties in the modeling and control design of a biological process using nonlinear models lead to the search for new and more efficient tools for both model-ing and control design. In this context, hybrid systems, i.e., systems includmodel-ing both continuous and discrete dynamics, open several possibilities for both modeling and control design. Chapter 4 is related with a modeling class of hybrid systems, viz., piecewise-linear (PWL) systems. The PWL approximation, i.e., systems which are linear or affine on each of the components of a polyhedral partition of the state space [96], have shown advantages on implementation, performance analysis, and calcula-tions [39], [84], [90], [89].

The problem to find a piecewise-linear model given a nonlinear model has been previously treated ([96], [39], [84], [90], and some others), in specific biological sys-tems [43]. More recently, an approximation for modeling gene-regulatory networks is presented in [13]. However, these approaches present many parameters that need to be provided by the designer, and finding these parameters is a difficult task, even for simple systems. In this work, a canonical piecewise linear approximation over simpli-cial partitions is used. It provides a partition of the state space into polytopic cells based on value at vertices [42], [89], [38]. This choice is motivated by several facts.

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First, this class of functions uniformly approximate any continuous nonlinear function

defined over a compact domain Rn (see [42]). Moreover, the canonical expression

in-troduced in [42] uses the minimum and exact number of parameters, and it is the first PWL expression able to represent PWL mappings in arbitrary dimensional domains. As a consequence of this, an efficient characterization is obtained from the viewpoint of memory storage and numerical evaluation [26]. Second, the approximation can be used in real implementations; the points taken from the nonlinear model may be replaced by points taken from sensors or data directly from the process. Thus, it addresses the problem of finding a PWL approximation of system where a reasonable number of measure samples of the vector field is available (regression set) [98]. Third, this alternative approach deals with an approximation which is easier to handle than the nonlinear model. In fact, it might use many tools developed for hybrid systems –e.g., the MLD model based approach [16]– since algorithms for translating MLD systems into PWL systems are available [15], [102]. Finally, this CPWL is used in a model based control, termed probing control in [68], being a first step development a hybrid probing control. The task of the controller is to determine, at every instant, the best control action (the best feed substrate) based on the compilation of the sensor’s on-line information (or for the nonlinear model). The fact that the probing strategy for feedback control requires a minimum of process knowledge is exploited in [7]. This work refers to a probing control as it is presented in [7] for E. coli. Short pulses to the feed rate are added, and taking into account the system response, the pulse is increased or decreased according with the tuning rule. The probing control strategy avoids acetate accumulation while maintaining a high growth rate [7], [101]. The main contribution of Chapter 4 is a hybrid dynamical model using

orthonor-mal high-level canonical piecewise linear functions [67], [66]. The approximation

model is tested by a recently presented control methodology, viz., the probing control strategy, which was developed in [7] for E. coli cultivations. It is implemented in

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simulations for this mathematical model [68]. The comparative analysis and error approximation between this new biological model and a nonlinear model developed first in [65], [64] are shown. This method is satisfactory for implementation purposes of a hybrid probing control [68].

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CHAPTER II

A POLYNOMIAL APPROACH FOR STABILITY

ANALYSIS OF SWITCHED SYSTEMS

The stability analysis of switched non-linear systems, i.e., continuous systems with switching signals under arbitrary switching, is treated in this chapter. Stability anal-ysis under arbitrary switching is a fundamental problem into the analanal-ysis and design of switched systems. For this problem, it is necessary that all the subsystems are asymptotically stable. However, in general, the above stability condition is not suffi-cient to guarantee stability for the switched system under arbitrary switching. It is well known that if there exists a common Lyapunov function for all the subsystems, then the stability of the switched system is guaranteed under arbitrary switching.

In this chapter we present a reformulation of the switched system as a differential continuous system on a constraint manifold. This representation opens several possi-bilities of analysis and design of switching systems in a consistent way, and also with numerical efficiency [69], [70], which are possible thanks to some tools developed in the last decade for polynomial differential-algebraic equations analysis [32], [75], [44]. Using this representation we develop an alternative method to search for a common Lyapunov function for switched systems with an efficient numerical method, using results for stability analysis of polynomial based on dissipativity theory [31], [70]. We propose a methodology to construct common Lyapunov functions for switched non-linear systems, which provides a less conservative test for proving stability under ar-bitrary switching. In Section 2.4, we extend the preliminary results to a more general, nonlinear case, and a representative example is presented to show the effectiveness of the methodology by reliable and efficient numerical methods. Basically, this theory

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is expressed in terms of an inequality involving a generalized system power input, or, supply rate, and a generalized energy function, or storage function [105]. The interpretation of this storage function establishes the connection between Lyapunov stability and dissipativity. Stability problems can be solved once the dissipativity property is assured, and also once the storage function becomes a Lyapunov function that can be used to construct Lyapunov functions for nonlinear dynamical systems.

2.1

Definitions and Preliminaries

2.1.1 Basic Concepts

A switched system is a system that consists of several continuous-time systems with discrete switching events. A switched system may be obtained from a hybrid system by neglecting the details of the discrete behavior and instead considering all possible switching patterns. Switched systems have many application, such as power electric circuits, automotive controllers, chemical processes, etc [54].

The mathematical model can be described by

˙x(t) = fσ(t)(x, t), (1)

where the state x ∈ Rn, fi : Rn× R+ → Rn are vector fields, and σ(t) : [0, tf]

Q ∈ {0, 1, ..., q} is a piecewise constant measurable function of time. Every mode of

operation corresponds to a specific subsystem ˙x(t) = fi(x, t), for some i∈ Q, and the

switching signal σ(t) determines which subsystem is active at each point in time on

the time interval [0, tf], with tf as the final time. No assumptions about the number of

switches or about the mode sequence are made. In addition, we consider a non-Zeno behavior, i.e., we exclude an infinite switching accumulation points in time. Finally, we assume that the state does not have jump discontinuities.

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−5 0 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 x1 x2 −5 0 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 x1 x2 −5 0 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 x1 x2

Figure 1: Switching between stable systems that are becoming stable

2.1.2 Stability Analysis under Arbitrary Switching and Dissipativity

The stability problem presents several interesting phenomena. For example, even when all the subsystems are exponentially stable, the switched system may be stable (see Figure 1) or may have divergent trajectories for certain switching signals (see Figure 2). Another scenario is also possible: one may carefully switch between un-stable subsystems to make the switched system exponentially un-stable [54]. We can see from these examples that the stability of switched systems depends not only on the dynamics of each subsystem but also on the properties of switching signals. There-fore, the stability study of switched systems can be roughly divided into two kinds of problems. One is the stability analysis of switched systems under given switching signals (maybe arbitrary, slow switching, etc.); the other is the synthesis of stabilizing switching signals for a given collection of dynamical systems. We are here dealing with the stability analysis of switched systems under arbitrary switching, i.e., the

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−5 0 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 x1 x2 −5 0 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 x1 x2 −5 0 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 x1 x2

Figure 2: Switching between stable systems that are becoming unstable

switched system state goes to zero asymptotically for any switching sequence. If this holds for any initial conditions, we have global uniform asymptotic stability (GUAS) [54], [56]. For this problem, it is necessary that all the subsystems be asymptotically stable. However, in general, the above subsystem stability assumption is not suffi-cient to assure stability for the switched systems under arbitrary switching, except for some special cases. On the other hand, if there exists a common Lyapunov function for all the subsystems, then the stability of the switched system is guaranteed under arbitrary switching. This provides us with a possible way to solve this problem, and a lot of efforts have been focused on the common quadratic Lyapunov functions [56].

2.1.2.1 Common Lyapunov functions

We are interested in obtaining a Lyapunov condition for GUAS. We proceed using the classic Lyapunov formulation. Given a positive definite continuously differentiable

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of systems (1) if there exists a positive definite continuous function W :Rn→ R such that

∂V

∂xfi(x)≤ −W (x) ∀x, ∀i ∈ Q.

Theorem 1 If all systems in the family (1) share a radially unbounded common

Lyapunov function, then the switched system is GUAS.

Theorem 1 is well known and can be derived in the same way as the standard Lya-punov stability theorem [54]. The main point is that the rate of decrease of V along solutions is not affected by switching; hence asymptotic stability is uniform with respect to σ.

2.1.2.2 A converse Lyapunov Theorem

The question now arises whether the existence of a common Lyapunov function is a more severe requirement than GUAS. A negative answer to this question –and a justification for the common Lyapunov function approach– follows from the converse Lyapunov theorem for switched systems [58], [30], [54] which claims that the GUAS property of a switched system implies the existence of a common Lyapunov function.

Theorem 2 Assume that the switched system (1) is GUAS, the set{fi(x) : i∈ Q} is

bounded for each x, and the function (x, i)→ fi(x) is locally Lipschitz in x uniformly

over i. Then all systems in the family (1) share a radially unbounded smooth common Lyapunov function.

There is a useful result which we find convenient to state here as a corollary of Theorem 2. It can be shown that if the switched system is GUAS, then all convex combinations of the individuals subsystems from the family (1) must be globally asymptotically stable. These convex combinations are defined by the vector fields

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Corollary 3 Under the assumption of Theorem 1, for every α∈ [0, 1] and all p, q ∈

Q, the system

˙x = fp,q,α(x)

is globally asymptotically stable.

2.1.2.3 Dissipativity

A switched system expressed as a polynomial differential-algebraic system allows us to establish an alternative approach for stability analysis. But instead of searching for a common Lyapunov function in order to provide stability under arbitrary switch-ing usswitch-ing traditional techniques (e.g., searchswitch-ing a sswitch-ingle Lyapunov function whose derivative along solutions of all systems in the family (1)satisfies suitable inequali-ties), which usually are very restrictive techniques based on exhaustive algorithms, as it is mentioned in the introduction of this chapter; we look for a Lyapunov function using techniques developed for polynomial continuous systems. It means that we can find a common Lyapunov function using dissipativity inequalities as in [31], or study the stability of constrained dynamical systems [81]. With this reformulation, we are dealing with a polynomial differential system on a manifold. Basically, the stability problem of differential-algebraic systems is related to the problem of stability on manifolds, which are defined by the constraints in the system description. From the concept of dissipativity, it could be inferred that storage functions induced by dissipativity are possible Lyapunov functions that are candidate for stability anal-ysis. This implies that stability and stabilization problems can be solved once the dissipativity property is assured [113]. It is possible to show that if the system is expressed as a purely passive system, the origin is an asymptotically unfluctuating equilibrium point, and the storage function V turns into a Lyapunov function. The functionality of stability analysis using dissipativity is that this property is preserved under interconnection [112], [113].

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2.2

An Equivalent Polynomial Representation

A polynomial expression able to mimic the behavior of a switched system is developed using a new variable s, which works as a parameter. The starting point is to rewrite (1) as a continuous non-switched control system in its more general case. The approach followed here has had in spirit some counterpart of 0-1 programs (see for instance [47]).

First, we define a drift vector field F(x) :Rn → Rn

F(x) = [f0(x) f1(x) ... fq(x)], (2)

where fi(x, u), i ∈ Q, is the function for each subsystem of the switched systems

given in (1). Let L be the vector of Lagrange polynomial interpolation quotients [25] defined with the new variable s, i.e.,

L(s) = [l0(s) l1(s) ... lq(s)]T, (3) where lk(s) = q  i=0 i=k (s− i) (k− i). (4)

We define the set

Γ ={s ∈ R |Q(s) = 0}, where Q(s) is the constraint polynomial so that

Q(s) =

q



k=0

(s− k), (5)

which is used to constrain s to take only integer values of the original set Q. Notice

that this clearly implies that we cannot find a solution if the starting point does not belong to this set. Finally, the solution of this system may be interpreted as an explicit ordinary differential equation (ODE) on the manifold Γ. A related continuous polynomial system of the switched system (1) is constructed in the following theorem.

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Theorem 4 Consider a switched system of the form given in (1) with a drift vector

field as is given in (2). Then, there exists a unique polynomial state system with a polynomial state equation p(x, s) of degree q in s, with s∈ Γ as follows:

˙x = p(x, s) = F(x)L(s) =

q



k=0

fk(x)lk(s). (6)

This polynomial system is an equivalent polynomial representation of the switched system (1).

Proof. Given a set of q + 1 subsystems f0(x), f1(x), ..., fq(x), using the definition of

the interpolation polynomial in the Lagrange form [25], we obtain a linear combination of the Lagrange basis polynomials as follows:

p(x, j) = fj(x), j = 0, 1, ..., q.

We use the Lagrange quotients that have the properties that lk(s) is a polynomial

(with degree q + 1), and that

lk(s) = δks ⎧ ⎪ ⎨ ⎪ ⎩ 1, s = k 0, s= k ,

where δksis the Dirac Delta function supported in ks. With this property, the function

p(x, s) can be defined as a polynomial in s with degree at most q, and

p(x, j) =

q



k=0

fk(x)lk(j) = fj(x).

There can be only one solution to the interpolation problem, since the difference of two solutions is a polynomial with degree at most q, and q + 1 zeros. This is only possible if the difference is identically zero, so p(x, s) is the unique polynomial interpolating the given set of subsystems. From the numerator in Equation (4), we

see that lk(s) is a polynomial of order q having zeros in all subsystems except the

k-th ones. The denominator is simply the constant that normalizes its value to 1 at k.

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Let q + 1 be the finite number of subsystems of the switched system (1), i.e.,

f0(x), . . . , fq(x). Then, the polynomial state equation p(x, s) is unique because the

quotients of the Lagrange polynomial interpolation l0, ..., lq are unique. Moreover, the

solutions of the algebraic equations Q(s) constrain the values of the variable s to be

in the set of finite values of the original setQ. Therefore, for any values of the s ∈ Γ,

the polynomial p(x, s) is equivalent to the switched system (1).

For instance, the most simple case arises when q = 1. In this case, the system (6) has the same form of the convex combination of two subsystems. When q = 2, the polynomial equivalent representation has the form

˙x = p(x, s) =2k=0fk(x)lk(s)

= 1

2f0(x)(s− 1)(s − 2) + f1(x)(s)(2− s) +

1

2f2(x)(s)(s− 1).

Notice that the trajectories of the original switched system (1) correspond to piecewise

constant controls taking values in the set σ ∈ {0, 1, ..., q}.

2.3

Results in Stability Analysis for Polynomial

Con-strained Dynamical Systems

In the previous section, the switched systems are expressed as polynomial differential-algebraic systems or constrained control systems. With this reformulation, we can apply the approach presented recently for constrained polynomial control systems based on dissipation inequalities [32], which is in spirit similar to the approach pre-sented in [81] (however the latter does not consider dissipation inequalities in its analysis). We can show that, with some assumptions, both approaches are equivalent from a computational point of view.

The main idea behind the proposed approach is to include in the system analysis the set of constraints, which are represented in this case by the semi-algebraic set Γ. Note that the semi-algebraic set Γ is equivalent to taking s as a constrained parameter that takes values on the roots of the polynomial Q(s). We need to check negative

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semidefinitness of ˙V (x) with respect to the constrained set Γ. We use the idea of

penalization used in optimization theory with constraints. For that, we use a function

λ(x, s), which can be interpreted as a penalization function or a Lagrange multiplier.

This idea is based on some results presented in [32] for constrained control systems, where we can use the dissipation inequality concept using storage functions and supply

rates [105]. Therefore, a dissipation inequality has the form ˙V (x)≤ a(x, ˙x, s), where

a(·) is an arbitrary scalar-valued function [31]. In the classical point of view, V (x)

is considered as the stored energy in the control system, and a(·) as the energy rate

supplied into the control system [105]. Note that the stability of general differential-algebraic systems has only been recently presented as a dissipation inequality [31]. In this approach, we take this idea of constrained stability analysis to deal with singular constrained control systems [32].

The following stability theorem is a particular case of the general result presented in [32], and it is used to find a common Lyapunov function for the switched system (1) through the equivalent polynomial representation (6).

Theorem 5 The equilibrium point x = 0 of the equivalent polynomial representation

(6) of the switched system (1) is stable for any admissible input s(t), if there exist polynomial functions V : Rn → R, λ : Rn× Γ → R, and a constraint polynomial

Q(s) = 0 such that V (x) is positive definite in a neighborhood of the origin, and λ(x, s)≥ 0 in Rn× Γ and the dissipation inequality

∂V

∂xp(x, s)≤ Q

2(s)λ(x, s) (7)

is satisfied for some neighborhood of the origin.

Proof. If the dissipation inequality (7) is satisfied, then the inequality can be

inte-grated in the interval [0, T )

V (0)− V (T ) ≥ −

T

0

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V (0)− V (T ) ≥ 0.

We have used the fact that Q(s) = 0, and s∈ Γ. This implies that (∂V/∂x)p(x, s) ≤ 0,

for all t ≥ 0, in some neighborhood of the origin. We consider also that Q2(s) is

positive or zero for all the values of s, in order to check semi-definiteness of ˙V (x),

and hence to satisfy the inequality above, we make λ(x, s) positive. It follows from this Lyapunov inequality and the continuity of the trajectories x(t), that V is not increasing and therefore the equilibrium point x = 0 of the system (6) is stable. Due

to the equivalence presented in Theorem 1, the equilibrium point x∗ = 0 is also an

equilibrium point for the switched system (1) for any admissible s∈ Γ.

Notice that the Lyapunov function V (x) used in Theorem 2 only depends on the state, i.e., it is a common Lyapunov function under arbitrary switching [54].

Remark 6 If we are interested in establishing asymptotic stability instead of stability,

then (7) must be satisfied strictly for all nonzero x in some x-neighborhood, i.e.,

∂V

∂xp(x, s) < Q

2(s)λ(x, s).

In general, it is very difficult to search for a Lyapunov function V (x) and a func-tion λ(x, s) for practical problems. However, recently established methods based on semidefinite programming and sum of squares decomposition allow us to verify Lya-punov inequalities of the form (7) very efficiently in the case where Q(s), V (x), and

λ(x, s) are assumed to be polynomials [31]. Certainly, in our case all of these functions

are of polynomial nature. It is impossible to search over all functions V (x), λ(x, s). In this approach, it is assumed that V (x) and λ(x, s) are polynomials up to certain degrees. Now, we can define the dissipation inequalities for the polynomial represen-tation of the switched system. Since we are studying global uniform asymptotically stable systems (GUAS), it means that we are searching for a common Lyapunov func-tion regardless of the switching sequence. Therefore, if we try to prove global stability

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of the system (6), the following polynomial inequalities must be satisfied,

V (x) > 0, and

∂V

∂xp(x, s)≤ Q

2(s)λ(x, s),

for all x ∈ Rn and s ∈ R. Note that if V (x) is polynomial and positive definite, it

implies that it is radially unbounded. To verify such polynomial inequalities is an NP-hard computational problem [31]. However, with the help of the sum of squares decomposition, it is possible to verify such polynomial inequalities very efficiently. On the other hand, this problem coincides with the problem of searching for a common

Lyapunov function for the vector field F(x) = [f0(x) f1(x) ... fq(x)].

For illustration and clarity of exposition, consider the case when q = 1. The dissipation inequality (7) becomes

∂V

∂x (f0(x)(1− s) + f1(x)s)≤ (s(s − 1))

2λ(x, s).

Before we state further results, we need to introduce some basic concepts of sum of squares decomposition. A more detailed description can be found in [77] and references therein.

2.3.1 The Sum of Squares Decomposition

In what follows, we present some basic concepts of the sum of squares decomposition technique to be used in the system analysis. The sum of squares decomposition is a method to check if a polynomial can be decomposed into a sum of squared polynomials.

Definition 7 [77] For x ∈ Rn, a multivariate polynomial p(x) is sum of squares

(SOS), if there exist some polynomials ri(x), i = 1, ..., M , such that

p(x) =

M



i=1

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It is clear that p(x) being SOS naturally implies p(x) ≥ 0, for all x ∈ Rn. An equivalent characterization of SOS polynomials is given in the following proposition taken from [77].

Proposition 8 [77] A polynomial p(x) of degree 2d is an SOS if and only if there

exists a positive semidefinite matrix Q and a vector of monomials Z(x) containing monomials in x of degree ≤ d such that

p(x) = Z(x)TQZ(x).

Since we have that p(x, s) is a polynomial vector field, and that we are searching for

V (x) that is also a polynomial in x, to solve the testing conditions inequality (7), we

can restrict our attention to cases in which the conditions admit SOS decompositions. The only apparent difficulty is the restriction of V (x) to be positive definite, not just positive semidefinite. To deal with this problem we can use the following proposition taken from [77].

Proposition 9 [77] Given a polynomial V (x) of degree 2d, let ϕ(x) =

n i=1 d j=1i,jx 2j i such that, d  j=1 i,j > γ ∀i = 1, ..., n, (9)

with γ a positive number, and i,j ≥ 0 for all i and j. Then the condition that

V (x)− ϕ(x) is SOS (10)

guarantees the positive definiteness of V (x).

Using these ideas, we can rewrite inequality (7), and a relaxation of Theorem 2 is stated in the following proposition.

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Proposition 10 For the equivalent polynomial representation system (6), if there

exist polynomial functions V (x), λ(x, s), and a positive definite function ϕ(x) of the form given in Proposition 6 such that

V (x)− ϕ(x) is SOS −∂V

∂xp(x, s) + Q2(s)λ(x, s) is SOS,

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then the polynomials V (x), λ(x, s), and the positive definite function ϕ(x) can be computed using SOSTOOLS [76].

The proof follows the same reasoning as in [77]. Therefore, Proposition 7 shows that with the polynomial equivalent representation in (5), we can obtain a common Lyapunov function using numerical tools. This Lyapunov function will be used to prove stability of the switched system (1).

Remark 11 Note that in Equation (11) the polynomials are sum of squares in terms

of x and s.

2.3.2 Numerical Example of a Polynomial Switched System

We present an illustrative example of a switched nonlinear system reformulated by Theorem 1 as an ordinary differential equation on a manifold. With this example we illustrate an efficient computational treatment to study stability analysis of switched systems using Theorem 2. Consider the set of systems described by the drift vector field F(x) = [f0(x) f1(x)], with f0(x) = ⎡ ⎢ ⎣ −β0x1+ x 2 1+ x22− α0x31 −β0x2+ 2x1x2− α0x32 ⎤ ⎥ ⎦ , and f1(x) = ⎡ ⎢ ⎣ −β1x1+ x 2 1+ x22− α1x31 −β1x2+ 2x1x2− α1x32 ⎤ ⎥ ⎦ ,

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This system is considered as a homogeneous switched system presented for stability analysis in [53]. In order to prove stability under arbitrary switching, we use the polynomial equivalent representation obtained using Theorem 1

˙x(t) = ⎛ ⎜ ⎝ f0(x) + ( ¯βx1+ ¯αx 3 1)s f0(x) + ( ¯βx2+ ¯αx3 2)s ⎞ ⎟ ⎠ , s ∈ Γ = {s ∈ R |Q(s) = s(s − 1) = 0},

with ¯β = (β1− β0) and ¯α = (α1 − α0). In [53] it is shown that βi > 2 and αi > 4

for i = 1, 2 in order to obtain a set of stable subsystems. We then set β0 = 10 and

α0 = 5 and, to keep ¯β > 2 and ¯α > 4, we set β1 = 13 and α1 = 10. We have obtained

a representation of the original system with a polynomial form, so that we can use Proposition 10 to analyze stability. First, we search for a Lyapunov function of the

polynomial form V (x) =  i,jai,jxixj. We have tried a function of degree 2 and

4, the latter corresponding to the function that we are looking for. With a degree of

2d = 4, and n = 2, we use a function ϕ(x) = 11x2

1+ 12x41+· · · + 22x42, where the ij

are the unknowns to be found by the tool, with a γ = 0.1, which implies ij ≥ γ.

For the penalty function, we have assumed a polynomial function of the same degree of the Lyapunov candidate function V (x), but considering also the s variable, i.e.,

λ(x, s) = b11x2

1 + b12x41 + b21x22 +· · · + b31s2 + b32s4. The coefficients bij are again

the unknown variables to be found. Using these polynomials and Equation (11), we obtain, using the MATLAB toolbox SOSTOOLS [76], a Lyapunov function of fourth degree, i.e.,

V (x) = 0.3x21+ 0.3944x22+ 0.11· 10−3x41+ 0.11· 10−3x42+ 0.7· 10−3x1x2

, which, through Theorem 2, proves that (0, 0) of the homogeneous nonlinear switched system, reformulated as a polynomial DAE (6), is a stable equilibrium point. Note that there is not a specific procedure to set the value of the degree of V (x) and the minimum value for γ (it should be noticed also that these functions are not

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unique). These parameters are chosen through different attempts. We start trying with a degree d = 1 and a small value of γ, and then we increase the degree until the properties of the Proposition 7 are met, and hence, we obtain a Lyapunov function. It may be interesting to develop an automatic pre-treatment algorithm to choose these values.

2.4

A Generalization for Nonlinear Switched Systems

In the previous sections we have focused our attention on switched systems of polyno-mial form, i.e., each subsystem is modeled by a polynopolyno-mial system. In this section we will extend the results to a more general class of switched systems, those modeled by elementary and nested elementary functions. This class of functions is related with explicit symbolic derivatives, such as exponential, logarithm, power-law, trigonomet-ric, and hyperbolic functions. For this aim, we transform, using a recasting process, the system obtained by the equivalent representation in a system with polynomial form, and then we use the results of Section 4 for stability analysis.

2.4.1 The Recasting Process for Stability Analysis

We use a recasting process introduced in [92], and later used for stability analysis of nonlinear systems in [75]. It is a procedure that goes through several steps until the system has the expected form. The algorithm is as follows:

• Step 0. Equivalent Representation: We consider the equivalent

represen-tation for the switched system obtained in Theorem 1, and we name it as the

original system (i.e., before the recasting process), with ξ = (ξ1, . . . , ξn) as the

state of the original system.

• Step 1. Original State Equations: The original system is described by

˙ ξi = j aj k pijk(ξ, s), i = 1, . . . , n. (12)

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Here ajs are real numbers, and the factors pijk are elementary functions, or nested elementary functions of elementary functions.

• Step 2. Decomposition of Non-Polynomial Functions: Let xi = ξi,

for i = 1, . . . , n. For each pijk(ξ, s) in Equation (12) that is not already a

power-law function, replace it with a new variable xn+1. This variable simplifies

the differential equation to sums and products of power-law functions. An

additional differential equation is generated for each new variable, using the chain rule of differentiation.

• Step 3. Recasting Process: When the recasting process leads to some constraints in the new variables, we have to introduce an n-dimensional manifold on which the solutions to the original differential equation lie. The particular choice of initial conditions defines the reference manifold.

• Step 4. The Polynomial Form: If the set of equations is in polynomial

form, then the recasting process is complete. If not, repeat steps 2-3 until to obtain a system of equations with a rational or polynomial form.

Remark 12 Notice that the constraints introduced by the definition of new variables,

and their initial conditions, restrict the system behavior to a manifold of the same dimension of the original problem.

As a result of the recasting process we have obtained new variables, which are consid-ered. Suppose that for a switched system consisting of subsystems of non-polynomial form, we apply the equivalent representation and obtain a system,

˙

ξ = p(ξ, s).

The recasted system obtained using the procedure presented above is written as ˙xo = po(xo, xr, s),

˙xr = pr(xo, xr, s),

Figure

Figure 1: Switching between stable systems that are becoming stable 2.1.2 Stability Analysis under Arbitrary Switching and Dissipativity The stability problem presents several interesting phenomena
Figure 2: Switching between stable systems that are becoming unstable switched system state goes to zero asymptotically for any switching sequence
Figure 3: Phase plane for two different initial conditions
Figure 4: States, co-states, and switching signal for the Arstein’s circle example
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